A Decentralized Quasi-Newton Method for Dual Formulations of Consensus Optimization
This addresses the problem of inefficient first-order methods in poorly conditioned consensus optimization for decentralized systems, though it is incremental as it builds on existing dual decomposition and BFGS techniques.
The paper tackles consensus optimization problems in decentralized networks by developing a dual D-BFGS method that incorporates curvature correction to improve conditioning, resulting in performance advantages over alternative algorithms as shown numerically.
This paper considers consensus optimization problems where each node of a network has access to a different summand of an aggregate cost function. Nodes try to minimize the aggregate cost function, while they exchange information only with their neighbors. We modify the dual decomposition method to incorporate a curvature correction inspired by the Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton method. The resulting dual D-BFGS method is a fully decentralized algorithm in which nodes approximate curvature information of themselves and their neighbors through the satisfaction of a secant condition. Dual D-BFGS is of interest in consensus optimization problems that are not well conditioned, making first order decentralized methods ineffective, and in which second order information is not readily available, making decentralized second order methods infeasible. Asynchronous implementation is discussed and convergence of D-BFGS is established formally for both synchronous and asynchronous implementations. Performance advantages relative to alternative decentralized algorithms are shown numerically.